ME DYNAMICAL SYSTEMS SPRING SEMESTER 2009

Transcription

1 ME DYNAMICAL SYSTEMS SPRING SEMESTER 2009 INSTRUCTOR Alfred Clark, Jr., Hopeman 329, x54078; clark@me.rochester.edu Office Hours: M T W Th F COURSE TIME AND PLACE T Th Hylan 306 PREREQUISITES The prerequisites are linear algebra (particularly the eigenvalue problem) and basic ordinary differential equations, including matrix methods for systems of equations. Because the linear algebra is so important, we will spend some time reviewing it at appropriate points in the course. In addition to linear algebra and ordinary differential equations, some prior experience with more advanced applied mathematics would be helpful -- for example, a course such as ME201/MTH 281. It would also be helpful (but not essential) to have some knowledge of elementary vibration theory. If you have any concerns about your background, I would be happy to discuss them with you. COURSE DESCRIPTION This course is an applied mathematics course designed to be an introduction to dynamical systems. Dynamical systems is a very large subject which includes such topics as nonlinear differential equations, iterated mappings, and fractals. Our course will concentrate on the nonlinear differential equations. The purpose of the course is to study the following question: What are the possible events in a system governed by nonlinear ordinary differential equations? The coverage will be more by example than by theorem. A major part of the course will be hands-on experience in exploring equations with the computer. The hardware and software to be used is described in more detail below. ASSIGNMENTS There will be 11 written homework assignments, most of which will involve computer work. There will be a three-hour mid-term exam covering the first half of the course on Wednesday April 1 (time to be arranged). The final exam, with emphasis on the second half of the course, is on Friday May 8, The grading weights are homework 50%, mid-term 25%, and final exam 25%.

2 ME 406 PAGE 2 OUTLINE OF COURSE The outline below is subject to changes as we proceed through the course. In addition to the topics listed, there will be applications at appropriate points in the course. On the last page of this handout, there is a detailed schedule of classes. We may deviate from this schedule as we go through the semester. An updated schedule will always be available online. 1. Plane Autonomous Systems Phase plane plots; stability of equilibrium for linear systems; stability for nonlinear systems by linearization; Liapunov's method for stability; periodic solutions; stability of periodic solutions; global phase portraits; bifurcations. 2. Higher Order Autonomous Systems Review of linear algebra and matrix methods for linear systems; local behavior near equilibrium points; Floquet theory and stability of periodic solutions; volume in phase space; general features of Lorenz equations; transition to chaos for Lorenz equations; fully developed chaos for Lorenz equations; tent map and the Lorenz equations; Liapunov exponents. 3. Selected Additional Topics (if time permits) The driven pendulum; dynamics of infectious diseases; the logistic map. TEXTBOOK The required text is Nonlinear Dynamics and Chaos, Steven H. Strogatz, Addison- Wesley, 1994, paperback edition Although we will cover about two-thirds of the material in the book, we will not follow it in detail in class lectures. It is an excellent text practical, authoritative and very readable. REFERENCES The literature in this field is extensive and varied in difficulty. The books on the short list below have been selected to be especially useful for broad coverage at the basic level of ME 406. The ones with an (R) designation at the end of the reference provide essential background for our course and are on reserve in Carlson. More specific references on special topics will be given during the course. Popular Treatments of Dynamical Systems and Chaos Does God Play Dice?, I. Stewart, 2 nd edition,blackwell, A superb overview of the field by a mathematician who has worked in the area. (R) Chaos, J. Gleick, Viking, 1987, available in paperback. Less technical than Stewart's book, but still providing a very nice overview.

3 ME 406 PAGE 3 The Essence of Chaos, E.N. Lorenz, University of Washington Press, Very interesting discussions of chaos, including some history of Lorenz s own discoveries. Background in Differential Equations Differential Equations, P. Blanchard, R.L. Devaney, and G.R. Hall, Brooks/Cole This outstanding text has been developed for a first course in differential equations, approaching the subject via systems of equations. It has very clear and basic discussions of many of the topics we cover, at a more elementary level than our course. (R) Ordinary Differential Equations, G. Birkhoff and G.-C. Rota, 4 th edition, Wiley This is an intermediate and excellent general text on ordinary differential equations. Chapter 5 is a good introduction to plane autonomous systems. (R) Theory of Ordinary Differential Equations, E. A. Coddington and N. Levinson, McGraw-Hill, 1955, Krieger reprint, Very advanced and comprehensive treatment, including a number of topics of importance in dynamical systems theory. General Treatments of Nonlinear Differential Equations and Dynamical Systems Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2 nd edition, M.W. Hirsch and S. Smale, and R.L. Devaney, Elsevier, This is a new edition of the elegant 1974 text by Hirsch and Smale. The most useful references for our course are our text, this book, and the books below by Jordan and Smith and Meiss. (R) Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, D.W. Jordan and P. Smith, 4 th edition, Oxford Press, This book is very readable, has broad coverage, and many excellent examples and problems. (R) Differential Dynamical Systems, James Meiss, SIAM, This recent applied math book was one of the possible texts for our course. You will find very helpful treatments in here of most of the topics covered in our course (and much more). (R) Differential Equations and Dynamical Systems, L. Perko, Springer-Verlag, 3 rd edition, This excellent book is written for mathematicians, but the background required is basic, and the book is not, for our purposes, overly abstract. If you want a treatment which is mathematically precise but accessible, this is the book to read. Another strength of the book is its organization, and the fact that it focuses completely on the fundamentals. The four chapters in the book are (1) Linear Systems, (2) Nonlinear Systems: Local Theory, (3) Nonlinear Systems: Global Theory, and (4) Nonlinear Systems: Bifurcation Theory. (R) Nonlinear Differential Equations and Dynamical Systems, F. Verhulst, Springer- Verlag, 2 nd edition, 1996, paperback. This book has more of a pure math flavor then Jordan and Smith, but it is still at an introductory level. It has less on chaos than we will cover. It is concise but very readable and has many examples. (R)

4 ME 406 PAGE 4 Chaotic Dynamics, G.L. Baker and J.P. Gollub, Cambridge, second edition, 1996, paperback. This excellent book is very readable. It deals with both discrete and differential systems, and it has a useful chapter on chaotic attractors. The book has much material on the driven pendulum, an important topic which, unfortunately, we probably will not have time to cover in our course. (R) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, J. Guckenheimer and P. Holmes, Springer-Verlag, This excellent book is the most accessible of the advanced books. If you are bothered by the vague nature of the discussion in many of the elementary books (or in our course!), you may find this book more satisfying, but be prepared to work hard when you pick it up. (R) Applications of Dynamical Systems Theory The text by Strogatz has many interesting applications. The references below collectively give a wide range of additional applications. Mathematical Biology, 3 rd edition, J.D. Murray, Springer-Verlag, Volume 1, 2002, and Volume II, Many of the biological applications we will cover are treated in this book, along with a host of other interesting problems. This is an indispensable reference for anyone interested in mathematical biology. (R) Mathematics for Dynamic Modeling, 2 nd edition, E. Beltrami, Academic Press, This book has a clear and elementary treatment of the basics of dynamical systems. It contains many interesting examples, such as algae blooms, the flywheel governor, the pumping heart, and the earth s magnetic field. (R) Chaotic Vibrations: An Introduction for Applied Scientists and Engineers, F. Moon, Wiley, This is an updated version of Moon s earlier books Chaotic Vibrations (1987) and Chaotic and Fractal Dynamics (1992). The book has a strong emphasis on physical applications, including a chapter on experimental studies of chaos. (R) COMPUTING TOOLS FOR THIS COURSE In some ways, this course is like an experimental course in which the laboratory is the computer. Although the class lectures will cover the basic mathematical theory, much of your understanding will come from pictures that you generate with the computer. In order that you may spend your time with the computer in exploration rather than in programming, you will be given a software package called DynPac with which to do the calculations and construct the graphs. DynPac runs in Mathematica. To help you get familiar with Mathematica, you will be given a basic Mathematica tutorial also. Both the tutorial and DynPac will be on the course CD handed out on the first day of class. If you wish to run this software on your own computer, you will need to have Mathematica installed on your machine. An alternative is to use the

5 ME 406 PAGE 5 Mathematica available on the Windows machines in the ME computer lab in the basement of Hopeman, room 05. The access to the lab is by combination lock, and the lab is available whenever the Hopeman building is open. VERSIONS OF MATHEMATICA At the present time, three successive versions of Mathematica are in general use: Mathematica 5 (usually version 5.2), Mathematica 6 (released in May of 2007), and Mathematica 7 (released last month). The official version for this course is Mathematica 6. All of the handouts and homework solutions will be in that version. There are a large number of minor incompatibilities between versions 5 and 6. Most notebooks written for one will not run in the other without modifications. My limited experience with version 7 suggests that there are few if any incompatibilities between versions 6 and 7. On the course disk you will find two sets of software: one set for Mathematica version 5, and one set for Mathematica version 6. (The version 6 software will almost certainly run in version 7 without any problems.) I would strongly encourage you to work in version 6 if possible. All of the machines in the ME computer lab are running version 6. COURSE WEB SITE The course web site will be the primary source of information about the course. Internet access is available on all of the machines in the ME Computer Lab. All examples presented in class will be available on the course web site. All homework assignments and solutions will be available there. Software used in the course will also be available there. It is important that you check the web site frequently, because any corrections or changes to homework or other important announcements will be posted there. Any updates to Dynpac will be posted there also. The URL for the web site is